find the domain and range of the following function. enter your answers in interval or set notation, as…

Explanation:

Step1: Recall domain of sine function

The sine function $y = \sin(u)$ is defined for all real - values of $u$. Here $u=\frac{\pi}{3}x + 5$, and $x\in R$.

Step2: Determine the domain of $f(x)$

Since there are no restrictions on $x$ for the function $f(x)=-5\sin(\frac{\pi}{3}x + 5)+4$, the domain is all real numbers. In interval notation, the domain is $(-\infty,\infty)$.

Step3: Recall range of sine function

The range of the basic sine function $y = \sin(u)$ is $[- 1,1]$, i.e., $-1\leqslant\sin(u)\leqslant1$.

Step4: Find the range of $-5\sin(u)$

Multiply the inequality $-1\leqslant\sin(u)\leqslant1$ by $- 5$. When we multiply an inequality by a negative number, the direction of the inequality signs changes. So we get $5\geqslant - 5\sin(u)\geqslant - 5$.

Step5: Find the range of $-5\sin(u)+4$

Add 4 to each part of the inequality $5\geqslant - 5\sin(u)\geqslant - 5$. We have $5 + 4\geqslant-5\sin(u)+4\geqslant-5 + 4$, which simplifies to $9\geqslant-5\sin(\frac{\pi}{3}x + 5)+4\geqslant - 1$.

Answer:

a) Domain: $(-\infty,\infty)$ b) Range: $[-1,9]$